Geodesic laminations

For a more detailed treatment see Casson (1983), or Harer–Penner (1986) or Chapter 8 of Thurston (1979).

Definition. Let S be a connected complete hyperbolic surface. Then a geodesic lamination on S is a closed subset λ of S which is a disjoint union of simple geodesics of S (which are called leaves of the lamination).

Remark: We allow the empty set as a geodesic lamination.

We denote the set of all geodesic laminations on S by Gℒ(S).

Definition. If λ is a lamination on S, then a component of S – λ is called a flat piece or a complementary region.

In general these need not be simply connected and may have a finite or infinite number of sides. The leaves of the geodesic lamination which form the boundary of some complementary region are called boundary leaves. For a surface of finite area there is an upper bound on the number of complementary regions, since each has area nπ for some positive integer n. Moreover each complementary region has finite type. On a surface with finite area, the set of boundary leaves is dense in the geodesic lamination.

Definition. A lamination such that each complementary region is isometric to an ideal triangle is said to be maximal.

We shall see, from Theorem I.4.2.8 (Structure of lamination) that any lamination on a surface of finite area can be extended by adding a finite number of new leaves to obtain a maximal lamination.